@Noam After a second thought ... I think I'm not surprised if someone can prove that $N(d)=\lfloor 2^d\delta(d)\rfloor$.

@S. Carnahan It's related to the hard but well studied densest packing problem. Of course I do not expect an "answer", but ideas are welcome. I updated the question.

@Noam, I'm using the best known result from Conway and Sloane's book. Since the sphere packing in dimension 8 and 24 are well understood, N(8) and N(24) should not be difficult. But I believe that, in general, the upperbound $\lfloor 2^d\delta(d)\rfloor$ is not optimal. A lattice packing achieving $\delta(d)$ satisfy some periodic boundary condition, but not the one defined by the unit cube.